On the computational complexity of the patrol boat scheduling problem with complete coverage

Abstract

Our study is primarily concerned with analyzing the computational complexity of the patrol boat scheduling problem with complete coverage (PBSPCC). This combinatorial optimization problem has important implications for maritime border protection and surveillance operations. The objective of the PBSPCC is to find a minimum size patrol boat fleet to provide ongoing continuous coverage at a set of maritime patrol regions, ensuring that there is at least one vessel on station in each patrol region at any given time. This requirement is complicated by the necessity for patrol vessels to be replenished on a regular basis in order to carry out patrol operations indefinitely. We introduce the PBSPCC via an example, discuss its relationship to related but dissimilar problems in the literature and proffer a mathematical description of the problem. We then show that the PBSPCC is  -hard by a transformation of the Hamiltonian graph decision problem into the problem of finding a minimum cyclic covering of a patrol network. We conclude that the associated decision problem of whether a patrol network has a continuous cover is  -complete, subject to the requirement that patrol covering solutions are cyclic of a bounded polynomial order.

Optimal Routing and Scheduling of Flag State Control Officers in Maritime Transportation

“…Applications for the PSRP can be widely found in different areas. Regarding the scheduling and routing problem for the security personnel, rounds of visits are conducted to various customer premises over a 24 h period in different locations, often using vehicles to travel between locations and then walking within the facilities [24][25][26]. In the healthcare domain, the home care problem entails local authorities offering community care services to residents.…”

Placement of Optical Sensors in 3D Terrain Using a Bacterial Evolutionary Algorithm

“…There are static, dynamic solutions for boundary surveillance. Dynamic solutions include different patrol mechanisms [ 25 , 26 ]. Generally, short—a few hours—flight times are typical for drones [ 27 ], but developments aiming at flight times of several hours [ 28 ] also exist.…”

On integer linear programming formulations of a patrol boat scheduling problem with complete coverage requirements

“…Surendonk and Chircop 2 have recently shown that the PBSPCC belongs firmly within the non-deterministic polynomial-time (NP)-hard computational complexity class. In contrast, related problems in the literature, such as the Maximum Covering and Patrol Routing Problem (MCPRP), 3,4 have been shown to exhibit only polynomial-time worst case complexity.…”

“…This is consistent with the NP-hard classification established by Surendonk and Chircop. 2 It has been observed that when using an objective function that seeks to minimize the total number of vessels, the column generation procedure is slow to produce good quality candidate patrol boat schedules. In order to speed-up the solution time, this paper proves a number of conditions under which an alternative objective, namely, minimize the total non-patrol time , can be used to arrive at a solution of the original problem.…”